Russian Physics Journal, Vol. 52, No. 2, 2009
FORMATION OF A PENTAGONAL PARTICLE STRUCTURE FROM COPPER NANOCLUSTERS A. G. Lipnitskii,1 D. N. Maradudin,1 D. N. Klimenko,1 E. V. Golosov,1 I. V. Nelasov,1 Yu. R. Kolobov,1 and A. A. Vikarchuk2
UDC 537.9
The structure of pentagonal particles and the processes of their formation from nanoclusters with the fifthorder symmetry axes are investigated by the methods of computer modeling and scanning electron-ion microscopy using copper as an example. It is demonstrated that the mechanism of cluster growth to pentagonal particles can be realized at which the volumetric stress present in noncrystal clusters will be released without breaking of the fifth-order symmetry of the growing cluster shape. Keywords: nanoclusters, pentagonal particles, copper, modeling. INTRODUCTION By the present time, isolated pentagonal particles of the majority of FCC metals have been prepared by various crystallization methods [1]. The particle shapes have the fifth-order axes, and the particle sizes are a few micrometers. For the first time, the researchers succeeded in obtaining the greatest variety of particle types and coatings and films comprising them by the method of electrodeposition of metals from solutions. Despite intensive investigations, the structure of pentagonal particles, mechanism of their growth, and special features of forming their properties caused by the fifth-order symmetry are widely debated topics [1]. In the present work, a mechanism of forming the pentagonal particles from nanoclusters is first investigated by computer modeling on the atomic level. The work is based on the approach described in [2]. The energy of copper nanoclusters shaped as perfect and imperfect decahedrons as well as perfect and imperfect pentagonal nanorods are compared for nanoclusters comprising from 103 to 106 atoms. As a result of modeling, a mechanism of pentagonal particle growth is suggested that allows volumetric stresses of the nanoclusters to be released retaining the fifth-order symmetry axes of their shapes. Results of modeling are confirmed by investigations of the pentagonal copper rod structure with a Quanta 200 3D scanning electron-ion microscope.
1. GEOMETRY AND CHARACTERISTICS OF THE NANOCLUSTERS The decahedron cluster structure is described in detail in the literature (for example, see [2] and references therein). Here we briefly describe the characteristics of the atomic cluster structure used below to perform computer modeling of copper clusters and to describe its results. Figure 1a shows a perfect decahedron copper cluster. In this case, the atomic cluster structure is described by the number of atoms m on the edge of each tetrahedron and by the number of atoms N in the cluster. The decahedron is formed by five tetrahedrons bounded by the {111} crystallographic planes. The tetrahedrons have the edge in common lying on the fifth-order decahedron axis. Each tetrahedron is adjacent to two other tetrahedrons and has the (111) face in common with each neighbor. The remaining tetrahedron faces form the decahedron surface which comprises ten {111}
1
Belgorod State University, Belgorod, Russia; 2Tol’yatti State University, Tol’yatti, Russia, e-mail:
[email protected];
[email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 27–32, February, 2009. Original article submitted June 11, 2008. 138
1064-8887/09/5202-0138 ©2009 Springer Science+Business Media, Inc.
Fig. 1. Perfect decahedron including N = 609 copper atoms formed by five deformed tetrahedrons with orientation of the common edge along the [110] crystallographic direction. The edge of each tetrahedron comprises m = 9 atoms, the common edge lies on the fifth-order decahedron axis (a); perfect nanorod with l = 10 and m = 4 (b); cut of a pentagonal copper rod prepared by electrodeposition (c); cross section of the relaxed decahedron cluster comprising N = 141774 atoms with built-in close-packed planes (parallel to the [110] direction) near one twin (d); cross section of the relaxed decahedron cluster comprising N = 921503 atoms with symmetrically built-in closepacked planes of atoms near each twin (e); cross section of the nanorod relaxed cluster comprising Nс = 10289 columns of atoms with built-in close-packed planes of atoms that fill two gaps between truncated tetrahedrons (f).
low-energy faces providing a low surface energy of the decahedron cluster. However, the ideal tetrahedron is enclosed in a solid angle of 70.5° (more precisely, arccos(1/3)); therefore, tetrahedrons must be deformed to fill the decahedron with five tetrahedrons without gaps to compensate for an angle of 7.5° required to obtain 360°. This leads to distortions of the FCC lattice and excessive volume energy, which results in energetically favorable crystal structure without volumetric deformations with increase in the cluster size. The pentagonal rod (Fig. 1b) can be considered as a decahedron in which each tetrahedron is truncated by the (100)-type plane parallel to the fifth-order decahedron axis (for more detailed consideration of the pentagonal nanorod structure, see [3] and the references therein). Taking into account the high ratio of the rod length to its transverse size, we neglect the end faces in calculations of the nanorod energy. This allows periodic boundary conditions to be used along the rod axis. The nanorods are formed by columns of atoms oriented along the rod axis (the [110] close-packed direction). For the examined rod, there are Nс columns comprising l atoms each, so that the number of atoms in the calculation grid is N = l × Nс. For this definition, the quantity l is equal to the period of the calculation grid along the rod axis in units of the shortest spacing of atoms, and Nс is the number of atoms in the plane perpendicular to the rod axis. The cross sectional rod area is proportional to Nс, and the rod radius is proportional to Nс1/2. We note that a continuous decahedron can also be constructed without deformation of tetrahedrons. For this purpose, it is suffice to build additional close-packed planes of atoms in the gaps between tetrahedrons. This will transform the atomic decahedron cluster structure into the crystal structure without excessive volumetric energy with
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five intergranular boundaries. As demonstrated the results of this work, exactly the transition to such structures from noncrystal ones with increase in their sizes explains the formation of pentagonal particle structure from growing clusters. The parameter convenient for a comparison of the stability of clusters having different sizes is given by the expression [2] Δ N = [ Eb ( N ) − N εcoh ]/ N 2/ 3 , where εcoh is the cohesive binding energy per atom in the single crystal,
Eb ( N ) is the binding energy of the cluster of N atoms. Thus, Δ N is the excess cluster energy compared to the perfect single crystal divided by the number of surface atoms proportional to N 2 / 3 . The excess cluster energy Δ N can be used to compare directly two clusters of the same sizes based on their excess energies. In addition, the quantity Δ N is also useful for a comparison of clusters having different sizes and for a description of contributions from the surface, jogs, and volume to the surface cluster energy. Thus, for clusters having the same shapes but different sizes, Δ N has the form [2]
Δ N = [cN 1/ 3 + bN 2 / 3 + aN ]/ N 2 / 3 .
(1)
Here the parameters c, b, and a describe linear, surface, and volumetric contributions to the surface cluster energy, respectively. The last contribution is present only for clusters with noncrystal structure. For nanorods, analogous characteristics have the form [3]
Δc = [ Eb ( N ) − N εcoh ]/(lNc1/ 2 ) and
(2)
Δ c = [c + bN c1/ 2
+ aNc ]/
Nc1/ 2
.
In this case, Δ c is the excess nanorod energy compared to the perfect single crystal divided by the number of surface atoms proportional to lNc1/ 2 for the nanorod.
2. CLUSTER MODELS Each cluster considered in this work was constructed in two stages. First, the initial arrangement of atoms in the cluster corresponding to a concrete structure type was assigned. Then relaxation to 0 K was carried out by the method of molecular dynamics. The relaxation terminated when the maximal force acting on each atom was
